Printable

2023 AMC 10A Problem 1

2023 AMC 10A Problem 1AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 1 min · position P1-10

Cities $A$ and $B$ are $45$ miles apart. Alicia lives in $A$ and Beth lives in $B$ . Alicia bikes towards $B$ at $18$ miles per hour. Leaving at the same time, Beth bikes toward $A$ at $12$ miles per hour. How many miles from City $A$ will they be when they meet?

A)
$20$
B)
$24$
C)
$25$
D)
$26$
E)
$27$

Answer

Key insight

Riders approaching each other close the gap at the combined speed 30 mph, so they meet after 1.5 hours; Alicia covers 18 times 1.5.

Solution

Since the two riders move toward each other, the distance between them shrinks at $18 + 12 = 30$ miles per hour. The $45$ -mile gap closes in
$\frac{45}{30} = 1.5 \text{ hours}.$

In that time Alicia, who starts at $A$ , rides $18 \cdot 1.5 = 27$ miles. So they meet $27$ miles from $A$ .

Check: Beth rides $12 \cdot 1.5 = 18$ miles, and $27 + 18 = 45$ .

The answer is $\boxed{\textbf{(E)}\ 27}$ .

Why this works

Two objects moving toward each other behave like one object moving at the sum of the speeds. Once the meeting time is known, each rider's distance is just rate times time. The final step is to answer the question actually asked: distance from $A$ , which is Alicia's distance.

Alternative approach

The distances are split in the ratio of the speeds, $18 : 12 = 3 : 2$ . Alicia covers $\frac{3}{5}$ of $45$ , which is $27$ .

The trap

Reporting Beth's distance (18 miles from B) or the meeting time instead of Alicia's distance from A.

Common mistakes

Reporting Beth's distance (18 miles from B) or the meeting time instead of Alicia's distance from A.
Using only Alicia's speed, $45 / 18 = 2.5$ hours, as if Beth were standing still.

Techniques

Set up the equation/formula and compute; no special trick needed